Two-stage submodular maximization problem beyond nonnegative and monotone

2021 ◽  
pp. 1-16
Author(s):  
Zhicheng Liu ◽  
Hong Chang ◽  
Ran Ma ◽  
Donglei Du ◽  
Xiaoyan Zhang
Keyword(s):  
Approximation Algorithms ◽  
Approximation Algorithm ◽  
Objective Function ◽  
Modular Function ◽  
Submodular Function ◽  
Maximization Problem ◽  
Cardinality Constraint ◽  
Two Stage ◽  
Time Efficiency ◽  
Submodular Maximization

Abstract We consider a two-stage submodular maximization problem subject to a cardinality constraint and k matroid constraints, where the objective function is the expected difference of a nonnegative monotone submodular function and a nonnegative monotone modular function. We give two bi-factor approximation algorithms for this problem. The first is a deterministic $\left( {{1 \over {k + 1}}\left( {1 - {1 \over {{e^{k + 1}}}}} \right),1} \right)$ -approximation algorithm, and the second is a randomized $\left( {{1 \over {k + 1}}\left( {1 - {1 \over {{e^{k + 1}}}}} \right) - \varepsilon ,1} \right)$ -approximation algorithm with improved time efficiency.

Monotone submodular maximization over the bounded integer lattice with cardinality constraints

2019 ◽  
Vol 11 (06) ◽  
pp. 1950075
Author(s):  
Lei Lai ◽  
Qiufen Ni ◽  
Changhong Lu ◽  
Chuanhe Huang ◽  
Weili Wu
Keyword(s):  
Approximation Algorithm ◽  
Greedy Algorithm ◽  
Deterministic Algorithm ◽  
Submodular Function ◽  
Integer Lattice ◽  
Cardinality Constraint ◽  
Cardinality Constraints ◽  
Definition Of ◽  
Deterministic Formula ◽  
Submodular Maximization

We consider the problem of maximizing monotone submodular function over the bounded integer lattice with a cardinality constraint. Function [Formula: see text] is submodular over integer lattice if [Formula: see text], [Formula: see text], where ∨ and ∧ represent elementwise maximum and minimum, respectively. Let [Formula: see text], and [Formula: see text], we study the problem of maximizing submodular function [Formula: see text] with constraints [Formula: see text] and [Formula: see text]. A random greedy [Formula: see text]-approximation algorithm and a deterministic [Formula: see text]-approximation algorithm are proposed in this paper. Both algorithms work in value oracle model. In the random greedy algorithm, we assume the monotone submodular function satisfies diminishing return property, which is not an equivalent definition of submodularity on integer lattice. Additionally, our random greedy algorithm makes [Formula: see text] value oracle queries and deterministic algorithm makes [Formula: see text] value oracle queries.

k-Submodular maximization with two kinds of constraints

2020 ◽  
pp. 2150036
Author(s):  
Ganquan Shi ◽  
Shuyang Gu ◽  
Weili Wu
Keyword(s):  
Approximation Algorithm ◽  
Submodular Function ◽  
Approximation Ratio ◽  
Size Difference ◽  
Total Size ◽  
Difference Constraints ◽  
Submodular Maximization ◽  
Individual Size

[Formula: see text]-submodular maximization is a generalization of submodular maximization, which requires us to select [Formula: see text] disjoint subsets instead of one subset. Attracted by practical values and applications, we consider [Formula: see text]-submodular maximization with two kinds of constraints. For total size and individual size difference constraints, we present a [Formula: see text]-approximation algorithm for maximizing a nonnegative k-submodular function, running in time [Formula: see text] at worst. Specially, if [Formula: see text] is multiple of [Formula: see text], the approximation ratio can reduce to [Formula: see text], running in time [Formula: see text] at worst. Besides, this algorithm can be applied to [Formula: see text]-bisubmodular achieving [Formula: see text]-approximation running in time [Formula: see text]. Furthermore, if [Formula: see text] is multiple of 2, the approximation ratio can reduce to [Formula: see text], running in time [Formula: see text] at worst. For individual size constraint, there is a [Formula: see text]-approximation algorithm for maximizing a nonnegative [Formula: see text]-submodular function and an nonnegative [Formula: see text]-bisubmodular function, running in time [Formula: see text] and [Formula: see text] respectively, at worst.

scholarly journals Optimal Bidding Strategy for Brand Advertising

2018 ◽  
Author(s):  
Takanori Maehara ◽  
Atsuhiro Narita ◽  
Jun Baba ◽  
Takayuki Kawabata
Keyword(s):  
Objective Function ◽  
Real World ◽  
Maximization Problem ◽  
Memory Retention ◽  
Real World Data ◽  
Real World Setting ◽  
Brand Advertising ◽  
Online Setting ◽  
Optimal Bidding ◽  
Submodular Maximization

Brand advertising is a type of advertising that aims at increasing the awareness of companies or products. This type of advertising is well studied in economic, marketing, and psychological literature; however, there are no studies in the area of computational advertising because the effect of such advertising is difficult to observe. In this study, we consider a real-time biding strategy for brand advertising. Here, our objective to maximizes the total number of users who remember the advertisement, averaged over the time. For this objective, we first introduce a new objective function that captures the cognitive psychological properties of memory retention, and can be optimized efficiently in the online setting (i.e., it is a monotone submodular function). Then, we propose an algorithm for the bid optimization problem with the proposed objective function under the second price mechanism by reducing the problem to the online knapsack constrained monotone submodular maximization problem. We evaluated the proposed objective function and the algorithm in a real-world data collected from our system and a questionnaire survey. We observed that our objective function is reasonable in real-world setting, and the proposed algorithm outperformed the baseline online algorithms.

Approximating Robust Parameterized Submodular Function Maximization in Large-Scales

2019 ◽  
Vol 36 (04) ◽  
pp. 1950022 ◽  
Author(s):  
Ruiqi Yang ◽  
Dachuan Xu ◽  
Yanjun Jiang ◽  
Yishui Wang ◽  
Dongmei Zhang
Keyword(s):  
Submodular Function ◽  
The Other ◽  
Selection Problem ◽  
Cardinality Constraint ◽  
Two Phase ◽  
Submodular Function Maximization ◽  
Set Function ◽  
Submodular Maximization

We study a robust parameterized submodular function maximization inspired by [Mitrović, S, I Bogunovic, A Norouzi-Fard and Jakub Tarnawski (2017). Streaming robust submodular maximization: A partitioned thresholding approach. In Proc. NIPS, pp. 4560–4569] and [Bogunovic, I, J Zhao and V Cevher (2018). Robust maximization of nonsubmodular objectives. In Proc. AISTATS, pp. 890–899]. In our setting, given a parameterized set function, there are two additional twists. One is that elements arrive in a streaming style, and the other is that there are at most [Formula: see text] items deleted from the algorithm’s memory when the stream is finished. The goal is to choose a robust set from the stream such that the robust ratio is maximized. We propose a two-phase algorithm for maximizing a normalized monotone robust parameterized submodular function with a cardinality constraint and show the robust ratio is close to a constant as [Formula: see text]. In the end, we empirically demonstrate the performance of our algorithm on deletion robust support selection problem.

scholarly journals Stochastic Submodular Maximization with Performance-Dependent Item Costs

2019 ◽  
Vol 33 ◽  
pp. 1485-1494
Author(s):  
Takuro Fukunaga ◽  
Takuya Konishi ◽  
Sumio Fujita ◽  
Ken-ichi Kawarabayashi
Keyword(s):  
Objective Function ◽  
Adaptive Algorithm ◽  
Numerical Experiments ◽  
Adaptive Algorithms ◽  
Budget Constraint ◽  
Maximization Problem ◽  
Maximum Value ◽  
Objective Value ◽  
The Cost ◽  
Submodular Maximization

We formulate a new stochastic submodular maximization problem by introducing the performance-dependent costs of items. In this problem, we consider selecting items for the case where the performance of each item (i.e., how much an item contributes to the objective function) is decided randomly, and the cost of an item depends on its performance. The goal of the problem is to maximize the objective function subject to a budget constraint on the costs of the selected items. We present an adaptive algorithm for this problem with a theoretical guaran-√ tee that its expected objective value is at least (1−1/ 4 e)/2 times the maximum value attained by any adaptive algorithms. We verify the performance of the algorithm through numerical experiments.

scholarly journals Top-k overlapping densest subgraphs: approximation algorithms and computational complexity

2020 ◽  
Author(s):  
Riccardo Dondi ◽  
Mohammad Mehdi Hosseinzadeh ◽  
Giancarlo Mauri ◽  
Italo Zoppis
Keyword(s):  
Computational Complexity ◽  
Approximation Algorithms ◽  
Approximation Algorithm ◽  
Objective Function ◽  
Graph Mining ◽  
Central Problem ◽  
Np Hard ◽  
Dense Subgraph ◽  
Dense Subgraphs ◽  
Approximation Factor

Abstract A central problem in graph mining is finding dense subgraphs, with several applications in different fields, a notable example being identifying communities. While a lot of effort has been put in the problem of finding a single dense subgraph, only recently the focus has been shifted to the problem of finding a set of densest subgraphs. An approach introduced to find possible overlapping subgraphs is the problem. Given an integer $$k \ge 1$$ k ≥ 1 and a parameter $$\lambda > 0$$ λ > 0 , the goal of this problem is to find a set of k dense subgraphs that may share some vertices. The objective function to be maximized takes into account the density of the subgraphs, the parameter $$\lambda $$ λ and the distance between each pair of subgraphs in the solution. The problem has been shown to admit a $$\frac{1}{10}$$ 1 10 -factor approximation algorithm. Furthermore, the computational complexity of the problem has been left open. In this paper, we present contributions concerning the approximability and the computational complexity of the problem. For the approximability, we present approximation algorithms that improve the approximation factor to $$\frac{1}{2}$$ 1 2 , when k is smaller than the number of vertices in the graph, and to $$\frac{2}{3}$$ 2 3 , when k is a constant. For the computational complexity, we show that the problem is NP-hard even when $$k=3$$ k = 3 .

Breaking the rmax Barrier: Enhanced Approximation Algorithms for Partial Set Multicover Problem

2020 ◽  
Author(s):  
Yingli Ran ◽  
Zhao Zhang ◽  
Shaojie Tang ◽  
Ding-Zhu Du
Keyword(s):  
Approximation Algorithms ◽  
Approximation Algorithm ◽  
Maximum Size ◽  
Fixed Number ◽  
Approximation Ratio ◽  
Cardinality Constraint ◽  
Minimum Density ◽  
The Cost ◽  
Element Set

Given an element set E of order n, a collection of subsets [Formula: see text], a cost cS on each set [Formula: see text], a covering requirement re for each element [Formula: see text], and an integer k, the goal of a minimum partial set multicover problem (MinPSMC) is to find a subcollection [Formula: see text] to fully cover at least k elements such that the cost of [Formula: see text] is as small as possible and element e is fully covered by [Formula: see text] if it belongs to at least re sets of [Formula: see text]. This problem generalizes the minimum k-union problem (MinkU) and is believed not to admit a subpolynomial approximation ratio. In this paper, we present a [Formula: see text]-approximation algorithm for MinPSMC, in which [Formula: see text] is the maximum size of a set in S. And when [Formula: see text], we present a bicriteria algorithm fully covering at least [Formula: see text] elements with approximation ratio [Formula: see text], where [Formula: see text] is a fixed number. These results are obtained by studying the minimum density subcollection problem with (or without) cardinality constraint, which might be of interest by itself.

scholarly journals Faster Guarantees of Evolutionary Algorithms for Maximization of Monotone Submodular Functions

2021 ◽  
Author(s):  
Victoria G. Crawford
Keyword(s):  
Optimization Algorithm ◽  
Empirical Evaluation ◽  
Pareto Optimization ◽  
Submodular Functions ◽  
Maximization Problem ◽  
Cardinality Constraint ◽  
Worst Case ◽  
Worst Case Ratio ◽  
Submodular Maximization ◽  
Greedy Solution

In this paper, the monotone submodular maximization problem (SM) is studied. SM is to find a subset of size kappa from a universe of size n that maximizes a monotone submodular objective function f . We show using a novel analysis that the Pareto optimization algorithm achieves a worst-case ratio of (1 − epsilon)(1 − 1/e) in expectation for every cardinality constraint kappa < P , where P ≤ n + 1 is an input, in O(nP ln(1/epsilon)) queries of f . In addition, a novel evolutionary algorithm called the biased Pareto optimization algorithm, is proposed that achieves a worst-case ratio of (1 − epsilon)(1 − 1/e − epsilon) in expectation for every cardinality constraint kappa < P in O(n ln(P ) ln(1/epsilon)) queries of f . Further, the biased Pareto optimization algorithm can be modified in order to achieve a a worst-case ratio of (1 − epsilon)(1 − 1/e − epsilon) in expectation for cardinality constraint kappa in O(n ln(1/epsilon)) queries of f . An empirical evaluation corroborates our theoretical analysis of the algorithms, as the algorithms exceed the stochastic greedy solution value at roughly when one would expect based upon our analysis.

Approximation Algorithms for Submodular Data Summarization with a Knapsack Constraint

2021 ◽  
Vol 5 (1) ◽  
pp. 1-31
Author(s):  
Kai Han ◽  
Shuang Cui ◽  
Tianshuai Zhu ◽  
Enpei Zhang ◽  
Benwei Wu ◽  
...  
Keyword(s):  
Approximation Algorithms ◽  
Fundamental Problem ◽  
Randomized Algorithm ◽  
Deterministic Algorithm ◽  
Submodular Function ◽  
Approximation Ratio ◽  
Performance Bounds ◽  
Data Summarization ◽  
Submodular Optimization ◽  
Knapsack Constraint

Data summarization, i.e., selecting representative subsets of manageable size out of massive data, is often modeled as a submodular optimization problem. Although there exist extensive algorithms for submodular optimization, many of them incur large computational overheads and hence are not suitable for mining big data. In this work, we consider the fundamental problem of (non-monotone) submodular function maximization with a knapsack constraint, and propose simple yet effective and efficient algorithms for it. Specifically, we propose a deterministic algorithm with approximation ratio 6 and a randomized algorithm with approximation ratio 4, and show that both of them can be accelerated to achieve nearly linear running time at the cost of weakening the approximation ratio by an additive factor of ε. We then consider a more restrictive setting without full access to the whole dataset, and propose streaming algorithms with approximation ratios of 8+ε and 6+ε that make one pass and two passes over the data stream, respectively. As a by-product, we also propose a two-pass streaming algorithm with an approximation ratio of 2+ε when the considered submodular function is monotone. To the best of our knowledge, our algorithms achieve the best performance bounds compared to the state-of-the-art approximation algorithms with efficient implementation for the same problem. Finally, we evaluate our algorithms in two concrete submodular data summarization applications for revenue maximization in social networks and image summarization, and the empirical results show that our algorithms outperform the existing ones in terms of both effectiveness and efficiency.

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